Personally I think we should draw a clear line between these two types of actions. When we don't do that, we weaken legitimate free speech.
Just to be very clear: I am in no way agreeing with Amazon, just correcting a confusion about the applicability of the US Constitution and amendments.
In other words, you are wrong to say "there is no free speech issue; the First Amendment protects the free speech from the government, nothing else". You would be right if you inserted "constitutional" or "legal" before "free speech issue".
The real substance of the issue is found in 'what kind of repercussions do you see as acceptable for exercising free speech', to which there are, in my opinion, many valid answers.